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// Three-dimensional graphing routines
private import math;
import graph;
import three;
typedef triple direction3(real);
direction3 Dir(triple dir) {return new triple(real) {return dir;};}
ticklocate ticklocate(real a, real b, autoscaleT S=defaultS,
real tickmin=-infinity, real tickmax=infinity,
real time(real)=null, direction3 dir)
{
if((valuetime) time == null) time=linear(S.T(),a,b);
ticklocate locate;
locate.a=a;
locate.b=b;
locate.S=S.copy();
if(finite(tickmin)) locate.S.tickMin=tickmin;
if(finite(tickmax)) locate.S.tickMax=tickmax;
locate.time=time;
locate.dir=zero;
locate.dir3=dir;
return locate;
}
private struct locateT {
real t; // tick location time
triple V; // tick location in frame coordinates
triple pathdir; // path direction in frame coordinates
triple dir; // tick direction in frame coordinates
void dir(transform3 T, path3 g, ticklocate locate, real t) {
pathdir=unit(shiftless(T)*dir(g,t));
triple Dir=locate.dir3(t);
dir=unit(Dir);
}
// Locate the desired position of a tick along a path.
void calc(transform3 T, path3 g, ticklocate locate, real val) {
t=locate.time(val);
V=T*point(g,t);
dir(T,g,locate,t);
}
}
void drawtick(picture pic, transform3 T, path3 g, path3 g2,
ticklocate locate, real val, real Size, int sign, pen p,
bool extend)
{
locateT locate1,locate2;
locate1.calc(T,g,locate,val);
path3 G;
if(extend && size(g2) > 0) {
locate2.calc(T,g2,locate,val);
G=locate1.V--locate2.V;
} else
G=(sign == 0) ?
locate1.V-Size*locate1.dir--locate1.V+Size*locate1.dir :
locate1.V--locate1.V+Size*sign*locate1.dir;
draw(pic,G,p,name="tick");
}
triple ticklabelshift(triple align, pen p=currentpen)
{
return 0.25*unit(align)*labelmargin(p);
}
// Signature of routines that draw labelled paths with ticks and tick labels.
typedef void ticks3(picture, transform3, Label, path3, path3, pen,
arrowbar3, margin3, ticklocate, int[], bool opposite=false,
bool primary=true);
// Label a tick on a frame.
void labeltick(picture pic, transform3 T, path3 g,
ticklocate locate, real val, int sign, real Size,
ticklabel ticklabel, Label F, real norm=0)
{
locateT locate1;
locate1.calc(T,g,locate,val);
triple align=F.align.dir3;
if(align == O) align=sign*locate1.dir;
triple shift=align*labelmargin(F.p);
if(dot(align,sign*locate1.dir) >= 0)
shift=sign*(Size)*locate1.dir;
real label;
if(locate.S.scale.logarithmic)
label=locate.S.scale.Tinv(val);
else {
label=val;
if(abs(label) < zerotickfuzz*norm) label=0;
// Fix epsilon errors at +/-1e-4
// default format changes to scientific notation here
if(abs(abs(label)-1e-4) < epsilon) label=sgn(label)*1e-4;
}
string s=ticklabel(label);
triple v=locate1.V+shift;
if(s != "")
label(pic,F.defaulttransform3 ? baseline(s,baselinetemplate) : F.T3*s,v,
align,F.p);
}
// Add axis label L to frame f.
void labelaxis(picture pic, transform3 T, Label L, path3 g,
ticklocate locate=null, int sign=1, bool ticklabels=false)
{
triple m=pic.min(identity4);
triple M=pic.max(identity4);
triple align=L.align.dir3;
Label L=L.copy();
pic.add(new void(frame f, transform3 T, picture pic2, projection P) {
path3 g=T*g;
real t=relative(L,g);
triple v=point(g,t);
picture F;
if(L.align.dir3 == O)
align=unit(invert(L.align.dir,v,P))*abs(L.align.dir);
if(ticklabels && locate != null && piecewisestraight(g)) {
locateT locate1;
locate1.dir(T,g,locate,t);
triple pathdir=locate1.pathdir;
triple perp=cross(pathdir,P.normal);
if(align == O)
align=unit(sgn(dot(sign*locate1.dir,perp))*perp);
path[] g=project(box(T*m,T*M),P);
pair z=project(v,P);
pair Ppathdir=project(v+pathdir,P)-z;
pair Perp=unit(I*Ppathdir);
real angle=degrees(Ppathdir,warn=false);
transform S=rotate(-angle,z);
path[] G=S*g;
pair Palign=project(v+align,P)-z;
pair Align=rotate(-angle)*dot(Palign,Perp)*Perp;
pair offset=unit(Palign)*
abs((Align.y >= 0 ? max(G).y : (Align.y < 0 ? min(G).y : 0))-z.y);
triple normal=cross(pathdir,align);
if(normal != O) v=invert(z+offset,normal,v,P);
}
label(F,L,v);
add(f,F.fit3(identity4,pic2,P));
},exact=false);
path3[] G=path3(texpath(L,bbox=true));
if(G.length > 0) {
G=L.align.is3D ? align(G,O,align,L.p) : L.T3*G;
triple v=point(g,relative(L,g));
pic.addBox(v,v,min(G),max(G));
}
}
// Tick construction routine for a user-specified array of tick values.
ticks3 Ticks3(int sign, Label F="", ticklabel ticklabel=null,
bool beginlabel=true, bool endlabel=true,
real[] Ticks=new real[], real[] ticks=new real[], int N=1,
bool begin=true, bool end=true,
real Size=0, real size=0, bool extend=false,
pen pTick=nullpen, pen ptick=nullpen)
{
return new void(picture pic, transform3 t, Label L, path3 g, path3 g2, pen p,
arrowbar3 arrow, margin3 margin, ticklocate locate,
int[] divisor, bool opposite, bool primary) {
// Use local copy of context variables:
int Sign=opposite ? -1 : 1;
int sign=Sign*sign;
pen pTick=pTick;
pen ptick=ptick;
ticklabel ticklabel=ticklabel;
real Size=Size;
real size=size;
if(Size == 0) Size=Ticksize;
if(size == 0) size=ticksize;
Label L=L.copy();
Label F=F.copy();
L.p(p);
F.p(p);
if(pTick == nullpen) pTick=p;
if(ptick == nullpen) ptick=pTick;
bool ticklabels=false;
path3 G=t*g;
path3 G2=t*g2;
scalefcn T;
real a,b;
if(locate.S.scale.logarithmic) {
a=locate.S.postscale.Tinv(locate.a);
b=locate.S.postscale.Tinv(locate.b);
T=locate.S.scale.T;
} else {
a=locate.S.Tinv(locate.a);
b=locate.S.Tinv(locate.b);
T=identity;
}
if(a > b) {real temp=a; a=b; b=temp;}
real norm=max(abs(a),abs(b));
string format=autoformat(F.s,norm...Ticks);
if(F.s == "%") F.s="";
if(ticklabel == null) {
if(locate.S.scale.logarithmic) {
int base=round(locate.S.scale.Tinv(1));
ticklabel=format == "%" ? Format("") : DefaultLogFormat(base);
} else ticklabel=Format(format);
}
bool labelaxis=L.s != "" && primary;
begingroup3(pic,"axis");
if(primary) draw(pic,margin(G,p).g,p,arrow);
else draw(pic,G,p);
for(int i=(begin ? 0 : 1); i < (end ? Ticks.length : Ticks.length-1); ++i) {
real val=T(Ticks[i]);
if(val >= a && val <= b)
drawtick(pic,t,g,g2,locate,val,Size,sign,pTick,extend);
}
for(int i=0; i < ticks.length; ++i) {
real val=T(ticks[i]);
if(val >= a && val <= b)
drawtick(pic,t,g,g2,locate,val,size,sign,ptick,extend);
}
if(N == 0) N=1;
if(Size > 0 && primary) {
for(int i=(beginlabel ? 0 : 1);
i < (endlabel ? Ticks.length : Ticks.length-1); i += N) {
real val=T(Ticks[i]);
if(val >= a && val <= b) {
ticklabels=true;
labeltick(pic,t,g,locate,val,Sign,Size,ticklabel,F,norm);
}
}
}
if(labelaxis)
labelaxis(pic,t,L,G,locate,Sign,ticklabels);
endgroup3(pic);
};
}
// Automatic tick construction routine.
ticks3 Ticks3(int sign, Label F="", ticklabel ticklabel=null,
bool beginlabel=true, bool endlabel=true,
int N, int n=0, real Step=0, real step=0,
bool begin=true, bool end=true, tickmodifier modify=None,
real Size=0, real size=0, bool extend=false,
pen pTick=nullpen, pen ptick=nullpen)
{
return new void(picture pic, transform3 T, Label L,
path3 g, path3 g2, pen p,
arrowbar3 arrow, margin3 margin=NoMargin3, ticklocate locate,
int[] divisor, bool opposite, bool primary) {
path3 G=T*g;
real limit=Step == 0 ? axiscoverage*arclength(G) : 0;
tickvalues values=modify(generateticks(sign,F,ticklabel,N,n,Step,step,
Size,size,identity(),1,
project(G,currentprojection),
limit,p,locate,divisor,
opposite));
Ticks3(sign,F,ticklabel,beginlabel,endlabel,values.major,values.minor,
values.N,begin,end,Size,size,extend,pTick,ptick)
(pic,T,L,g,g2,p,arrow,margin,locate,divisor,opposite,primary);
};
}
ticks3 NoTicks3()
{
return new void(picture pic, transform3 T, Label L, path3 g,
path3, pen p, arrowbar3 arrow, margin3 margin,
ticklocate, int[], bool opposite, bool primary) {
path3 G=T*g;
if(primary) draw(pic,margin(G,p).g,p,arrow,margin);
else draw(pic,G,p);
if(L.s != "" && primary) {
Label L=L.copy();
L.p(p);
labelaxis(pic,T,L,G,opposite ? -1 : 1);
}
};
}
ticks3 InTicks(Label format="", ticklabel ticklabel=null,
bool beginlabel=true, bool endlabel=true,
int N=0, int n=0, real Step=0, real step=0,
bool begin=true, bool end=true, tickmodifier modify=None,
real Size=0, real size=0, bool extend=false,
pen pTick=nullpen, pen ptick=nullpen)
{
return Ticks3(-1,format,ticklabel,beginlabel,endlabel,N,n,Step,step,
begin,end,modify,Size,size,extend,pTick,ptick);
}
ticks3 OutTicks(Label format="", ticklabel ticklabel=null,
bool beginlabel=true, bool endlabel=true,
int N=0, int n=0, real Step=0, real step=0,
bool begin=true, bool end=true, tickmodifier modify=None,
real Size=0, real size=0, bool extend=false,
pen pTick=nullpen, pen ptick=nullpen)
{
return Ticks3(1,format,ticklabel,beginlabel,endlabel,N,n,Step,step,
begin,end,modify,Size,size,extend,pTick,ptick);
}
ticks3 InOutTicks(Label format="", ticklabel ticklabel=null,
bool beginlabel=true, bool endlabel=true,
int N=0, int n=0, real Step=0, real step=0,
bool begin=true, bool end=true, tickmodifier modify=None,
real Size=0, real size=0, bool extend=false,
pen pTick=nullpen, pen ptick=nullpen)
{
return Ticks3(0,format,ticklabel,beginlabel,endlabel,N,n,Step,step,
begin,end,modify,Size,size,extend,pTick,ptick);
}
ticks3 InTicks(Label format="", ticklabel ticklabel=null,
bool beginlabel=true, bool endlabel=true,
real[] Ticks, real[] ticks=new real[],
real Size=0, real size=0, bool extend=false,
pen pTick=nullpen, pen ptick=nullpen)
{
return Ticks3(-1,format,ticklabel,beginlabel,endlabel,
Ticks,ticks,Size,size,extend,pTick,ptick);
}
ticks3 OutTicks(Label format="", ticklabel ticklabel=null,
bool beginlabel=true, bool endlabel=true,
real[] Ticks, real[] ticks=new real[],
real Size=0, real size=0, bool extend=false,
pen pTick=nullpen, pen ptick=nullpen)
{
return Ticks3(1,format,ticklabel,beginlabel,endlabel,
Ticks,ticks,Size,size,extend,pTick,ptick);
}
ticks3 InOutTicks(Label format="", ticklabel ticklabel=null,
bool beginlabel=true, bool endlabel=true,
real[] Ticks, real[] ticks=new real[],
real Size=0, real size=0, bool extend=false,
pen pTick=nullpen, pen ptick=nullpen)
{
return Ticks3(0,format,ticklabel,beginlabel,endlabel,
Ticks,ticks,Size,size,extend,pTick,ptick);
}
ticks3 NoTicks3=NoTicks3(),
InTicks=InTicks(),
OutTicks=OutTicks(),
InOutTicks=InOutTicks();
triple tickMin3(picture pic)
{
return minbound(pic.userMin(),(pic.scale.x.tickMin,pic.scale.y.tickMin,
pic.scale.z.tickMin));
}
triple tickMax3(picture pic)
{
return maxbound(pic.userMax(),(pic.scale.x.tickMax,pic.scale.y.tickMax,
pic.scale.z.tickMax));
}
axis Bounds(int type=Both, int type2=Both, triple align=O, bool extend=false)
{
return new void(picture pic, axisT axis) {
axis.type=type;
axis.type2=type2;
axis.position=0.5;
axis.align=align;
axis.extend=extend;
};
}
axis YZEquals(real y, real z, triple align=O, bool extend=false)
{
return new void(picture pic, axisT axis) {
axis.type=Value;
axis.type2=Value;
axis.value=pic.scale.y.T(y);
axis.value2=pic.scale.z.T(z);
axis.position=1;
axis.align=align;
axis.extend=extend;
};
}
axis XZEquals(real x, real z, triple align=O, bool extend=false)
{
return new void(picture pic, axisT axis) {
axis.type=Value;
axis.type2=Value;
axis.value=pic.scale.x.T(x);
axis.value2=pic.scale.z.T(z);
axis.position=1;
axis.align=align;
axis.extend=extend;
};
}
axis XYEquals(real x, real y, triple align=O, bool extend=false)
{
return new void(picture pic, axisT axis) {
axis.type=Value;
axis.type2=Value;
axis.value=pic.scale.x.T(x);
axis.value2=pic.scale.y.T(y);
axis.position=1;
axis.align=align;
axis.extend=extend;
};
}
axis YZZero(triple align=O, bool extend=false)
{
return new void(picture pic, axisT axis) {
axis.type=Value;
axis.type2=Value;
axis.value=pic.scale.y.T(pic.scale.y.scale.logarithmic ? 1 : 0);
axis.value2=pic.scale.z.T(pic.scale.z.scale.logarithmic ? 1 : 0);
axis.position=1;
axis.align=align;
axis.extend=extend;
};
}
axis XZZero(triple align=O, bool extend=false)
{
return new void(picture pic, axisT axis) {
axis.type=Value;
axis.type2=Value;
axis.value=pic.scale.x.T(pic.scale.x.scale.logarithmic ? 1 : 0);
axis.value2=pic.scale.z.T(pic.scale.z.scale.logarithmic ? 1 : 0);
axis.position=1;
axis.align=align;
axis.extend=extend;
};
}
axis XYZero(triple align=O, bool extend=false)
{
return new void(picture pic, axisT axis) {
axis.type=Value;
axis.type2=Value;
axis.value=pic.scale.x.T(pic.scale.x.scale.logarithmic ? 1 : 0);
axis.value2=pic.scale.y.T(pic.scale.y.scale.logarithmic ? 1 : 0);
axis.position=1;
axis.align=align;
axis.extend=extend;
};
}
axis
Bounds=Bounds(),
YZZero=YZZero(),
XZZero=XZZero(),
XYZero=XYZero();
// Draw a general three-dimensional axis.
void axis(picture pic=currentpicture, Label L="", path3 g, path3 g2=nullpath3,
pen p=currentpen, ticks3 ticks, ticklocate locate,
arrowbar3 arrow=None, margin3 margin=NoMargin3,
int[] divisor=new int[], bool above=false, bool opposite=false)
{
Label L=L.copy();
real t=reltime(g,0.5);
if(L.defaultposition) L.position(t);
divisor=copy(divisor);
locate=locate.copy();
pic.add(new void (picture f, transform3 t, transform3 T, triple, triple) {
picture d;
ticks(d,t,L,g,g2,p,arrow,margin,locate,divisor,opposite,true);
add(f,t*T*inverse(t)*d);
},above=above);
addPath(pic,g,p);
if(L.s != "") {
frame f;
Label L0=L.copy();
L0.position(0);
add(f,L0);
triple pos=point(g,L.relative()*length(g));
pic.addBox(pos,pos,min3(f),max3(f));
}
}
real xtrans(transform3 t, real x)
{
return (t*(x,0,0)).x;
}
real ytrans(transform3 t, real y)
{
return (t*(0,y,0)).y;
}
real ztrans(transform3 t, real z)
{
return (t*(0,0,z)).z;
}
private triple defaultdir(triple X, triple Y, triple Z, bool opposite=false,
projection P) {
triple u=cross(P.normal,Z);
return abs(dot(u,X)) > abs(dot(u,Y)) ? -X : (opposite ? Y : -Y);
}
// An internal routine to draw an x axis at a particular y value.
void xaxis3At(picture pic=currentpicture, Label L="", axis axis,
real xmin=-infinity, real xmax=infinity, pen p=currentpen,
ticks3 ticks=NoTicks3,
arrowbar3 arrow=None, margin3 margin=NoMargin3, bool above=true,
bool opposite=false, bool opposite2=false, bool primary=true)
{
int type=axis.type;
int type2=axis.type2;
triple dir=axis.align.dir3 == O ?
defaultdir(Y,Z,X,opposite^opposite2,currentprojection) : axis.align.dir3;
Label L=L.copy();
if(L.align.dir3 == O && L.align.dir == 0) L.align(opposite ? -dir : dir);
real y=axis.value;
real z=axis.value2;
real y2,z2;
int[] divisor=copy(axis.xdivisor);
pic.add(new void(picture f, transform3 t, transform3 T, triple lb,
triple rt) {
transform3 tinv=inverse(t);
triple a=xmin == -infinity ? tinv*(lb.x-min3(p).x,ytrans(t,y),
ztrans(t,z)) : (xmin,y,z);
triple b=xmax == infinity ? tinv*(rt.x-max3(p).x,ytrans(t,y),
ztrans(t,z)) : (xmax,y,z);
real y0;
real z0;
if(abs(dir.y) < abs(dir.z)) {
y0=y;
z0=z2;
} else {
y0=y2;
z0=z;
}
triple a2=xmin == -infinity ? tinv*(lb.x-min3(p).x,ytrans(t,y0),
ztrans(t,z0)) : (xmin,y0,z0);
triple b2=xmax == infinity ? tinv*(rt.x-max3(p).x,ytrans(t,y0),
ztrans(t,z0)) : (xmax,y0,z0);
if(xmin == -infinity || xmax == infinity) {
bounds mx=autoscale(a.x,b.x,pic.scale.x.scale);
pic.scale.x.tickMin=mx.min;
pic.scale.x.tickMax=mx.max;
divisor=mx.divisor;
}
triple fuzz=X*epsilon*max(abs(a.x),abs(b.x));
a -= fuzz;
b += fuzz;
picture d;
ticks(d,t,L,a--b,finite(y0) && finite(z0) ? a2--b2 : nullpath3,
p,arrow,margin,
ticklocate(a.x,b.x,pic.scale.x,Dir(dir)),divisor,
opposite,primary);
add(f,t*T*tinv*d);
},above=above);
void bounds() {
if(type == Min)
y=pic.scale.y.automin() ? tickMin3(pic).y : pic.userMin().y;
else if(type == Max)
y=pic.scale.y.automax() ? tickMax3(pic).y : pic.userMax().y;
else if(type == Both) {
y2=pic.scale.y.automax() ? tickMax3(pic).y : pic.userMax().y;
y=opposite ? y2 :
(pic.scale.y.automin() ? tickMin3(pic).y : pic.userMin().y);
}
if(type2 == Min)
z=pic.scale.z.automin() ? tickMin3(pic).z : pic.userMin().z;
else if(type2 == Max)
z=pic.scale.z.automax() ? tickMax3(pic).z : pic.userMax().z;
else if(type2 == Both) {
z2=pic.scale.z.automax() ? tickMax3(pic).z : pic.userMax().z;
z=opposite2 ? z2 :
(pic.scale.z.automin() ? tickMin3(pic).z : pic.userMin().z);
}
real Xmin=finite(xmin) ? xmin : pic.userMin().x;
real Xmax=finite(xmax) ? xmax : pic.userMax().x;
triple a=(Xmin,y,z);
triple b=(Xmax,y,z);
triple a2=(Xmin,y2,z2);
triple b2=(Xmax,y2,z2);
if(finite(a)) {
pic.addPoint(a,min3(p));
pic.addPoint(a,max3(p));
}
if(finite(b)) {
pic.addPoint(b,min3(p));
pic.addPoint(b,max3(p));
}
if(finite(a) && finite(b)) {
picture d;
ticks(d,pic.scaling3(warn=false),L,
(a.x,0,0)--(b.x,0,0),(a2.x,0,0)--(b2.x,0,0),p,arrow,margin,
ticklocate(a.x,b.x,pic.scale.x,Dir(dir)),divisor,
opposite,primary);
frame f;
if(L.s != "") {
Label L0=L.copy();
L0.position(0);
add(f,L0);
}
triple pos=a+L.relative()*(b-a);
triple m=min3(d);
triple M=max3(d);
pic.addBox(pos,pos,(min3(f).x,m.y,m.z),(max3(f).x,m.y,m.z));
}
}
// Process any queued y and z axes bound calculation requests.
for(int i=0; i < pic.scale.y.bound.length; ++i)
pic.scale.y.bound[i]();
for(int i=0; i < pic.scale.z.bound.length; ++i)
pic.scale.z.bound[i]();
pic.scale.y.bound.delete();
pic.scale.z.bound.delete();
bounds();
// Request another x bounds calculation before final picture scaling.
pic.scale.x.bound.push(bounds);
}
// An internal routine to draw an x axis at a particular y value.
void yaxis3At(picture pic=currentpicture, Label L="", axis axis,
real ymin=-infinity, real ymax=infinity, pen p=currentpen,
ticks3 ticks=NoTicks3,
arrowbar3 arrow=None, margin3 margin=NoMargin3, bool above=true,
bool opposite=false, bool opposite2=false, bool primary=true)
{
int type=axis.type;
int type2=axis.type2;
triple dir=axis.align.dir3 == O ?
defaultdir(X,Z,Y,opposite^opposite2,currentprojection) : axis.align.dir3;
Label L=L.copy();
if(L.align.dir3 == O && L.align.dir == 0) L.align(opposite ? -dir : dir);
real x=axis.value;
real z=axis.value2;
real x2,z2;
int[] divisor=copy(axis.ydivisor);
pic.add(new void(picture f, transform3 t, transform3 T, triple lb,
triple rt) {
transform3 tinv=inverse(t);
triple a=ymin == -infinity ? tinv*(xtrans(t,x),lb.y-min3(p).y,
ztrans(t,z)) : (x,ymin,z);
triple b=ymax == infinity ? tinv*(xtrans(t,x),rt.y-max3(p).y,
ztrans(t,z)) : (x,ymax,z);
real x0;
real z0;
if(abs(dir.x) < abs(dir.z)) {
x0=x;
z0=z2;
} else {
x0=x2;
z0=z;
}
triple a2=ymin == -infinity ? tinv*(xtrans(t,x0),lb.y-min3(p).y,
ztrans(t,z0)) : (x0,ymin,z0);
triple b2=ymax == infinity ? tinv*(xtrans(t,x0),rt.y-max3(p).y,
ztrans(t,z0)) : (x0,ymax,z0);
if(ymin == -infinity || ymax == infinity) {
bounds my=autoscale(a.y,b.y,pic.scale.y.scale);
pic.scale.y.tickMin=my.min;
pic.scale.y.tickMax=my.max;
divisor=my.divisor;
}
triple fuzz=Y*epsilon*max(abs(a.y),abs(b.y));
a -= fuzz;
b += fuzz;
picture d;
ticks(d,t,L,a--b,finite(x0) && finite(z0) ? a2--b2 : nullpath3,
p,arrow,margin,
ticklocate(a.y,b.y,pic.scale.y,Dir(dir)),divisor,
opposite,primary);
add(f,t*T*tinv*d);
},above=above);
void bounds() {
if(type == Min)
x=pic.scale.x.automin() ? tickMin3(pic).x : pic.userMin().x;
else if(type == Max)
x=pic.scale.x.automax() ? tickMax3(pic).x : pic.userMax().x;
else if(type == Both) {
x2=pic.scale.x.automax() ? tickMax3(pic).x : pic.userMax().x;
x=opposite ? x2 :
(pic.scale.x.automin() ? tickMin3(pic).x : pic.userMin().x);
}
if(type2 == Min)
z=pic.scale.z.automin() ? tickMin3(pic).z : pic.userMin().z;
else if(type2 == Max)
z=pic.scale.z.automax() ? tickMax3(pic).z : pic.userMax().z;
else if(type2 == Both) {
z2=pic.scale.z.automax() ? tickMax3(pic).z : pic.userMax().z;
z=opposite2 ? z2 :
(pic.scale.z.automin() ? tickMin3(pic).z : pic.userMin().z);
}
real Ymin=finite(ymin) ? ymin : pic.userMin().y;
real Ymax=finite(ymax) ? ymax : pic.userMax().y;
triple a=(x,Ymin,z);
triple b=(x,Ymax,z);
triple a2=(x2,Ymin,z2);
triple b2=(x2,Ymax,z2);
if(finite(a)) {
pic.addPoint(a,min3(p));
pic.addPoint(a,max3(p));
}
if(finite(b)) {
pic.addPoint(b,min3(p));
pic.addPoint(b,max3(p));
}
if(finite(a) && finite(b)) {
picture d;
ticks(d,pic.scaling3(warn=false),L,
(0,a.y,0)--(0,b.y,0),(0,a2.y,0)--(0,a2.y,0),p,arrow,margin,
ticklocate(a.y,b.y,pic.scale.y,Dir(dir)),divisor,
opposite,primary);
frame f;
if(L.s != "") {
Label L0=L.copy();
L0.position(0);
add(f,L0);
}
triple pos=a+L.relative()*(b-a);
triple m=min3(d);
triple M=max3(d);
pic.addBox(pos,pos,(m.x,min3(f).y,m.z),(m.x,max3(f).y,m.z));
}
}
// Process any queued x and z axis bound calculation requests.
for(int i=0; i < pic.scale.x.bound.length; ++i)
pic.scale.x.bound[i]();
for(int i=0; i < pic.scale.z.bound.length; ++i)
pic.scale.z.bound[i]();
pic.scale.x.bound.delete();
pic.scale.z.bound.delete();
bounds();
// Request another y bounds calculation before final picture scaling.
pic.scale.y.bound.push(bounds);
}
// An internal routine to draw an x axis at a particular y value.
void zaxis3At(picture pic=currentpicture, Label L="", axis axis,
real zmin=-infinity, real zmax=infinity, pen p=currentpen,
ticks3 ticks=NoTicks3,
arrowbar3 arrow=None, margin3 margin=NoMargin3, bool above=true,
bool opposite=false, bool opposite2=false, bool primary=true)
{
int type=axis.type;
int type2=axis.type2;
triple dir=axis.align.dir3 == O ?
defaultdir(X,Y,Z,opposite^opposite2,currentprojection) : axis.align.dir3;
Label L=L.copy();
if(L.align.dir3 == O && L.align.dir == 0) L.align(opposite ? -dir : dir);
real x=axis.value;
real y=axis.value2;
real x2,y2;
int[] divisor=copy(axis.zdivisor);
pic.add(new void(picture f, transform3 t, transform3 T, triple lb,
triple rt) {
transform3 tinv=inverse(t);
triple a=zmin == -infinity ? tinv*(xtrans(t,x),ytrans(t,y),
lb.z-min3(p).z) : (x,y,zmin);
triple b=zmax == infinity ? tinv*(xtrans(t,x),ytrans(t,y),
rt.z-max3(p).z) : (x,y,zmax);
real x0;
real y0;
if(abs(dir.x) < abs(dir.y)) {
x0=x;
y0=y2;
} else {
x0=x2;
y0=y;
}
triple a2=zmin == -infinity ? tinv*(xtrans(t,x0),ytrans(t,y0),
lb.z-min3(p).z) : (x0,y0,zmin);
triple b2=zmax == infinity ? tinv*(xtrans(t,x0),ytrans(t,y0),
rt.z-max3(p).z) : (x0,y0,zmax);
if(zmin == -infinity || zmax == infinity) {
bounds mz=autoscale(a.z,b.z,pic.scale.z.scale);
pic.scale.z.tickMin=mz.min;
pic.scale.z.tickMax=mz.max;
divisor=mz.divisor;
}
triple fuzz=Z*epsilon*max(abs(a.z),abs(b.z));
a -= fuzz;
b += fuzz;
picture d;
ticks(d,t,L,a--b,finite(x0) && finite(y0) ? a2--b2 : nullpath3,
p,arrow,margin,
ticklocate(a.z,b.z,pic.scale.z,Dir(dir)),divisor,
opposite,primary);
add(f,t*T*tinv*d);
},above=above);
void bounds() {
if(type == Min)
x=pic.scale.x.automin() ? tickMin3(pic).x : pic.userMin().x;
else if(type == Max)
x=pic.scale.x.automax() ? tickMax3(pic).x : pic.userMax().x;
else if(type == Both) {
x2=pic.scale.x.automax() ? tickMax3(pic).x : pic.userMax().x;
x=opposite ? x2 :
(pic.scale.x.automin() ? tickMin3(pic).x : pic.userMin().x);
}
if(type2 == Min)
y=pic.scale.y.automin() ? tickMin3(pic).y : pic.userMin().y;
else if(type2 == Max)
y=pic.scale.y.automax() ? tickMax3(pic).y : pic.userMax().y;
else if(type2 == Both) {
y2=pic.scale.y.automax() ? tickMax3(pic).y : pic.userMax().y;
y=opposite2 ? y2 :
(pic.scale.y.automin() ? tickMin3(pic).y : pic.userMin().y);
}
real Zmin=finite(zmin) ? zmin : pic.userMin().z;
real Zmax=finite(zmax) ? zmax : pic.userMax().z;
triple a=(x,y,Zmin);
triple b=(x,y,Zmax);
triple a2=(x2,y2,Zmin);
triple b2=(x2,y2,Zmax);
if(finite(a)) {
pic.addPoint(a,min3(p));
pic.addPoint(a,max3(p));
}
if(finite(b)) {
pic.addPoint(b,min3(p));
pic.addPoint(b,max3(p));
}
if(finite(a) && finite(b)) {
picture d;
ticks(d,pic.scaling3(warn=false),L,
(0,0,a.z)--(0,0,b.z),(0,0,a2.z)--(0,0,a2.z),p,arrow,margin,
ticklocate(a.z,b.z,pic.scale.z,Dir(dir)),divisor,
opposite,primary);
frame f;
if(L.s != "") {
Label L0=L.copy();
L0.position(0);
add(f,L0);
}
triple pos=a+L.relative()*(b-a);
triple m=min3(d);
triple M=max3(d);
pic.addBox(pos,pos,(m.x,m.y,min3(f).z),(m.x,m.y,max3(f).z));
}
}
// Process any queued x and y axes bound calculation requests.
for(int i=0; i < pic.scale.x.bound.length; ++i)
pic.scale.x.bound[i]();
for(int i=0; i < pic.scale.y.bound.length; ++i)
pic.scale.y.bound[i]();
pic.scale.x.bound.delete();
pic.scale.y.bound.delete();
bounds();
// Request another z bounds calculation before final picture scaling.
pic.scale.z.bound.push(bounds);
}
// Internal routine to autoscale the user limits of a picture.
void autoscale3(picture pic=currentpicture, axis axis)
{
bool set=pic.scale.set;
autoscale(pic,axis);
if(!set) {
bounds mz;
if(pic.userSetz()) {
mz=autoscale(pic.userMin().z,pic.userMax().z,pic.scale.z.scale);
if(pic.scale.z.scale.logarithmic &&
floor(pic.userMin().z) == floor(pic.userMax().z)) {
if(pic.scale.z.automin())
pic.userMinz3(floor(pic.userMin().z));
if(pic.scale.z.automax())
pic.userMaxz3(ceil(pic.userMax().z));
}
} else {mz.min=mz.max=0; pic.scale.set=false;}
pic.scale.z.tickMin=mz.min;
pic.scale.z.tickMax=mz.max;
axis.zdivisor=mz.divisor;
}
}
// Draw an x axis in three dimensions.
void xaxis3(picture pic=currentpicture, Label L="", axis axis=YZZero,
real xmin=-infinity, real xmax=infinity, pen p=currentpen,
ticks3 ticks=NoTicks3,
arrowbar3 arrow=None, margin3 margin=NoMargin3, bool above=false)
{
if(xmin > xmax) return;
if(pic.scale.x.automin && xmin > -infinity) pic.scale.x.automin=false;
if(pic.scale.x.automax && xmax < infinity) pic.scale.x.automax=false;
if(!pic.scale.set) {
axis(pic,axis);
autoscale3(pic,axis);
}
bool newticks=false;
if(xmin != -infinity) {
xmin=pic.scale.x.T(xmin);
newticks=true;
}
if(xmax != infinity) {
xmax=pic.scale.x.T(xmax);
newticks=true;
}
if(newticks && pic.userSetx() && ticks != NoTicks3) {
if(xmin == -infinity) xmin=pic.userMin().x;
if(xmax == infinity) xmax=pic.userMax().x;
bounds mx=autoscale(xmin,xmax,pic.scale.x.scale);
pic.scale.x.tickMin=mx.min;
pic.scale.x.tickMax=mx.max;
axis.xdivisor=mx.divisor;
}
axis(pic,axis);
if(xmin == -infinity && !axis.extend) {
if(pic.scale.set)
xmin=pic.scale.x.automin() ? pic.scale.x.tickMin :
max(pic.scale.x.tickMin,pic.userMin().x);
else xmin=pic.userMin().x;
}
if(xmax == infinity && !axis.extend) {
if(pic.scale.set)
xmax=pic.scale.x.automax() ? pic.scale.x.tickMax :
min(pic.scale.x.tickMax,pic.userMax().x);
else xmax=pic.userMax().x;
}
if(L.defaultposition) {
L=L.copy();
L.position(axis.position);
}
bool back=false;
if(axis.type == Both) {
triple v=currentprojection.normal;
back=dot((0,pic.userMax().y-pic.userMin().y,0),v)*sgn(v.z) > 0;
}
xaxis3At(pic,L,axis,xmin,xmax,p,ticks,arrow,margin,above,false,false,!back);
if(axis.type == Both)
xaxis3At(pic,L,axis,xmin,xmax,p,ticks,arrow,margin,above,true,false,back);
if(axis.type2 == Both) {
xaxis3At(pic,L,axis,xmin,xmax,p,ticks,arrow,margin,above,false,true,false);
if(axis.type == Both)
xaxis3At(pic,L,axis,xmin,xmax,p,ticks,arrow,margin,above,true,true,false);
}
}
// Draw a y axis in three dimensions.
void yaxis3(picture pic=currentpicture, Label L="", axis axis=XZZero,
real ymin=-infinity, real ymax=infinity, pen p=currentpen,
ticks3 ticks=NoTicks3,
arrowbar3 arrow=None, margin3 margin=NoMargin3, bool above=false)
{
if(ymin > ymax) return;
if(pic.scale.y.automin && ymin > -infinity) pic.scale.y.automin=false;
if(pic.scale.y.automax && ymax < infinity) pic.scale.y.automax=false;
if(!pic.scale.set) {
axis(pic,axis);
autoscale3(pic,axis);
}
bool newticks=false;
if(ymin != -infinity) {
ymin=pic.scale.y.T(ymin);
newticks=true;
}
if(ymax != infinity) {
ymax=pic.scale.y.T(ymax);
newticks=true;
}
if(newticks && pic.userSety() && ticks != NoTicks3) {
if(ymin == -infinity) ymin=pic.userMin().y;
if(ymax == infinity) ymax=pic.userMax().y;
bounds my=autoscale(ymin,ymax,pic.scale.y.scale);
pic.scale.y.tickMin=my.min;
pic.scale.y.tickMax=my.max;
axis.ydivisor=my.divisor;
}
axis(pic,axis);
if(ymin == -infinity && !axis.extend) {
if(pic.scale.set)
ymin=pic.scale.y.automin() ? pic.scale.y.tickMin :
max(pic.scale.y.tickMin,pic.userMin().y);
else ymin=pic.userMin().y;
}
if(ymax == infinity && !axis.extend) {
if(pic.scale.set)
ymax=pic.scale.y.automax() ? pic.scale.y.tickMax :
min(pic.scale.y.tickMax,pic.userMax().y);
else ymax=pic.userMax().y;
}
if(L.defaultposition) {
L=L.copy();
L.position(axis.position);
}
bool back=false;
if(axis.type == Both) {
triple v=currentprojection.normal;
back=dot((pic.userMax().x-pic.userMin().x,0,0),v)*sgn(v.z) > 0;
}
yaxis3At(pic,L,axis,ymin,ymax,p,ticks,arrow,margin,above,false,false,!back);
if(axis.type == Both)
yaxis3At(pic,L,axis,ymin,ymax,p,ticks,arrow,margin,above,true,false,back);
if(axis.type2 == Both) {
yaxis3At(pic,L,axis,ymin,ymax,p,ticks,arrow,margin,above,false,true,false);
if(axis.type == Both)
yaxis3At(pic,L,axis,ymin,ymax,p,ticks,arrow,margin,above,true,true,false);
}
}
// Draw a z axis in three dimensions.
void zaxis3(picture pic=currentpicture, Label L="", axis axis=XYZero,
real zmin=-infinity, real zmax=infinity, pen p=currentpen,
ticks3 ticks=NoTicks3,
arrowbar3 arrow=None, margin3 margin=NoMargin3, bool above=false)
{
if(zmin > zmax) return;
if(pic.scale.z.automin && zmin > -infinity) pic.scale.z.automin=false;
if(pic.scale.z.automax && zmax < infinity) pic.scale.z.automax=false;
if(!pic.scale.set) {
axis(pic,axis);
autoscale3(pic,axis);
}
bool newticks=false;
if(zmin != -infinity) {
zmin=pic.scale.z.T(zmin);
newticks=true;
}
if(zmax != infinity) {
zmax=pic.scale.z.T(zmax);
newticks=true;
}
if(newticks && pic.userSetz() && ticks != NoTicks3) {
if(zmin == -infinity) zmin=pic.userMin().z;
if(zmax == infinity) zmax=pic.userMax().z;
bounds mz=autoscale(zmin,zmax,pic.scale.z.scale);
pic.scale.z.tickMin=mz.min;
pic.scale.z.tickMax=mz.max;
axis.zdivisor=mz.divisor;
}
axis(pic,axis);
if(zmin == -infinity && !axis.extend) {
if(pic.scale.set)
zmin=pic.scale.z.automin() ? pic.scale.z.tickMin :
max(pic.scale.z.tickMin,pic.userMin().z);
else zmin=pic.userMin().z;
}
if(zmax == infinity && !axis.extend) {
if(pic.scale.set)
zmax=pic.scale.z.automax() ? pic.scale.z.tickMax :
min(pic.scale.z.tickMax,pic.userMax().z);
else zmax=pic.userMax().z;
}
if(L.defaultposition) {
L=L.copy();
L.position(axis.position);
}
bool back=false;
if(axis.type == Both) {
triple v=currentprojection.vector();
back=dot((pic.userMax().x-pic.userMin().x,0,0),v)*sgn(v.y) > 0;
}
zaxis3At(pic,L,axis,zmin,zmax,p,ticks,arrow,margin,above,false,false,!back);
if(axis.type == Both)
zaxis3At(pic,L,axis,zmin,zmax,p,ticks,arrow,margin,above,true,false,back);
if(axis.type2 == Both) {
zaxis3At(pic,L,axis,zmin,zmax,p,ticks,arrow,margin,above,false,true,false);
if(axis.type == Both)
zaxis3At(pic,L,axis,zmin,zmax,p,ticks,arrow,margin,above,true,true,false);
}
}
// Set the z limits of a picture.
void zlimits(picture pic=currentpicture, real min=-infinity, real max=infinity,
bool crop=NoCrop)
{
if(min > max) return;
pic.scale.z.automin=min <= -infinity;
pic.scale.z.automax=max >= infinity;
bounds mz;
if(pic.scale.z.automin() || pic.scale.z.automax())
mz=autoscale(pic.userMin().z,pic.userMax().z,pic.scale.z.scale);
if(pic.scale.z.automin) {
if(pic.scale.z.automin()) pic.userMinz(mz.min);
} else pic.userMinz(min(pic.scale.z.T(min),pic.scale.z.T(max)));
if(pic.scale.z.automax) {
if(pic.scale.z.automax()) pic.userMaxz(mz.max);
} else pic.userMaxz(max(pic.scale.z.T(min),pic.scale.z.T(max)));
}
// Restrict the x, y, and z limits to box(min,max).
void limits(picture pic=currentpicture, triple min, triple max)
{
xlimits(pic,min.x,max.x);
ylimits(pic,min.y,max.y);
zlimits(pic,min.z,max.z);
}
// Draw x, y and z axes.
void axes3(picture pic=currentpicture,
Label xlabel="", Label ylabel="", Label zlabel="",
bool extend=false,
triple min=(-infinity,-infinity,-infinity),
triple max=(infinity,infinity,infinity),
pen p=currentpen, arrowbar3 arrow=None, margin3 margin=NoMargin3)
{
xaxis3(pic,xlabel,YZZero(extend),min.x,max.x,p,arrow,margin);
yaxis3(pic,ylabel,XZZero(extend),min.y,max.y,p,arrow,margin);
zaxis3(pic,zlabel,XYZero(extend),min.z,max.z,p,arrow,margin);
}
triple Scale(picture pic=currentpicture, triple v)
{
return (pic.scale.x.T(v.x),pic.scale.y.T(v.y),pic.scale.z.T(v.z));
}
real ScaleZ(picture pic=currentpicture, real z)
{
return pic.scale.z.T(z);
}
// Draw a tick of length size at triple v in direction dir using pen p.
void tick(picture pic=currentpicture, triple v, triple dir, real size=Ticksize,
pen p=currentpen)
{
triple v=Scale(pic,v);
pic.add(new void (picture f, transform3 t) {
triple tv=t*v;
draw(f,tv--tv+unit(dir)*size,p);
});
pic.addPoint(v,p);
pic.addPoint(v,unit(dir)*size,p);
}
void xtick(picture pic=currentpicture, triple v, triple dir=Y,
real size=Ticksize, pen p=currentpen)
{
tick(pic,v,dir,size,p);
}
void xtick3(picture pic=currentpicture, real x, triple dir=Y,
real size=Ticksize, pen p=currentpen)
{
tick(pic,(x,pic.scale.y.scale.logarithmic ? 1 : 0,
pic.scale.z.scale.logarithmic ? 1 : 0),dir,size,p);
}
void ytick(picture pic=currentpicture, triple v, triple dir=X,
real size=Ticksize, pen p=currentpen)
{
tick(pic,v,dir,size,p);
}
void ytick3(picture pic=currentpicture, real y, triple dir=X,
real size=Ticksize, pen p=currentpen)
{
tick(pic,(pic.scale.x.scale.logarithmic ? 1 : 0,y,
pic.scale.z.scale.logarithmic ? 1 : 0),dir,size,p);
}
void ztick(picture pic=currentpicture, triple v, triple dir=X,
real size=Ticksize, pen p=currentpen)
{
xtick(pic,v,dir,size,p);
}
void ztick3(picture pic=currentpicture, real z, triple dir=X,
real size=Ticksize, pen p=currentpen)
{
xtick(pic,(pic.scale.x.scale.logarithmic ? 1 : 0,
pic.scale.y.scale.logarithmic ? 1 : 0,z),dir,size,p);
}
void tick(picture pic=currentpicture, Label L, real value, triple v,
triple dir, string format="", real size=Ticksize, pen p=currentpen)
{
Label L=L.copy();
L.align(L.align,-dir);
if(shift(L.T3)*O == O)
L.T3=shift(dot(dir,L.align.dir3) > 0 ? dir*size :
ticklabelshift(L.align.dir3,p))*L.T3;
L.p(p);
if(L.s == "") L.s=format(format == "" ? defaultformat : format,value);
L.s=baseline(L.s,baselinetemplate);
label(pic,L,Scale(pic,v));
tick(pic,v,dir,size,p);
}
void xtick(picture pic=currentpicture, Label L, triple v, triple dir=Y,
string format="", real size=Ticksize, pen p=currentpen)
{
tick(pic,L,v.x,v,dir,format,size,p);
}
void xtick3(picture pic=currentpicture, Label L, real x, triple dir=Y,
string format="", real size=Ticksize, pen p=currentpen)
{
xtick(pic,L,(x,pic.scale.y.scale.logarithmic ? 1 : 0,
pic.scale.z.scale.logarithmic ? 1 : 0),dir,size,p);
}
void ytick(picture pic=currentpicture, Label L, triple v, triple dir=X,
string format="", real size=Ticksize, pen p=currentpen)
{
tick(pic,L,v.y,v,dir,format,size,p);
}
void ytick3(picture pic=currentpicture, Label L, real y, triple dir=X,
string format="", real size=Ticksize, pen p=currentpen)
{
xtick(pic,L,(pic.scale.x.scale.logarithmic ? 1 : 0,y,
pic.scale.z.scale.logarithmic ? 1 : 0),dir,format,size,p);
}
void ztick(picture pic=currentpicture, Label L, triple v, triple dir=X,
string format="", real size=Ticksize, pen p=currentpen)
{
tick(pic,L,v.z,v,dir,format,size,p);
}
void ztick3(picture pic=currentpicture, Label L, real z, triple dir=X,
string format="", real size=Ticksize, pen p=currentpen)
{
xtick(pic,L,(pic.scale.x.scale.logarithmic ? 1 : 0,
pic.scale.z.scale.logarithmic ? 1 : 0,z),dir,format,size,p);
}
private void label(picture pic, Label L, triple v, real x, align align,
string format, pen p)
{
Label L=L.copy();
L.align(align);
L.p(p);
if(shift(L.T3)*O == O)
L.T3=shift(ticklabelshift(L.align.dir3,L.p))*L.T3;
if(L.s == "") L.s=format(format == "" ? defaultformat : format,x);
L.s=baseline(L.s,baselinetemplate);
label(pic,L,v);
}
void labelx(picture pic=currentpicture, Label L="", triple v,
align align=-Y, string format="", pen p=currentpen)
{
label(pic,L,Scale(pic,v),v.x,align,format,p);
}
void labelx3(picture pic=currentpicture, Label L="", real x,
align align=-Y, string format="", pen p=currentpen)
{
labelx(pic,L,(x,pic.scale.y.scale.logarithmic ? 1 : 0,
pic.scale.z.scale.logarithmic ? 1 : 0),align,format,p);
}
void labely(picture pic=currentpicture, Label L="", triple v,
align align=-X, string format="", pen p=currentpen)
{
label(pic,L,Scale(pic,v),v.y,align,format,p);
}
void labely3(picture pic=currentpicture, Label L="", real y,
align align=-X, string format="", pen p=currentpen)
{
labely(pic,L,(pic.scale.x.scale.logarithmic ? 1 : 0,y,
pic.scale.z.scale.logarithmic ? 1 : 0),align,format,p);
}
void labelz(picture pic=currentpicture, Label L="", triple v,
align align=-X, string format="", pen p=currentpen)
{
label(pic,L,Scale(pic,v),v.z,align,format,p);
}
void labelz3(picture pic=currentpicture, Label L="", real z,
align align=-X, string format="", pen p=currentpen)
{
labelz(pic,L,(pic.scale.x.scale.logarithmic ? 1 : 0,
pic.scale.y.scale.logarithmic ? 1 : 0,z),align,format,p);
}
typedef guide3 graph(triple F(real), real, real, int);
typedef guide3[] multigraph(triple F(real), real, real, int);
graph graph(interpolate3 join)
{
return new guide3(triple f(real), real a, real b, int n) {
real width=b-a;
return n == 0 ? join(f(a)) :
join(...sequence(new guide3(int i) {return f(a+(i/n)*width);},n+1));
};
}
multigraph graph(interpolate3 join, bool3 cond(real))
{
return new guide3[](triple f(real), real a, real b, int n) {
real width=b-a;
if(n == 0) return new guide3[] {join(cond(a) ? f(a) : nullpath3)};
guide3[] G;
guide3[] g;
for(int i=0; i < n+1; ++i) {
real t=a+(i/n)*width;
bool3 b=cond(t);
if(b)
g.push(f(t));
else {
if(g.length > 0) {
G.push(join(...g));
g=new guide3[] {};
}
if(b == default)
g.push(f(t));
}
}
if(g.length > 0)
G.push(join(...g));
return G;
};
}
guide3 Straight(... guide3[])=operator --;
guide3 Spline(... guide3[])=operator ..;
guide3 graph(picture pic=currentpicture, real x(real), real y(real),
real z(real), real a, real b, int n=ngraph,
interpolate3 join=operator --)
{
return graph(join)(new triple(real t) {return Scale(pic,(x(t),y(t),z(t)));},
a,b,n);
}
guide3[] graph(picture pic=currentpicture, real x(real), real y(real),
real z(real), real a, real b, int n=ngraph,
bool3 cond(real), interpolate3 join=operator --)
{
return graph(join,cond)(new triple(real t) {
return Scale(pic,(x(t),y(t),z(t)));
},a,b,n);
}
guide3 graph(picture pic=currentpicture, triple v(real), real a, real b,
int n=ngraph, interpolate3 join=operator --)
{
return graph(join)(new triple(real t) {return Scale(pic,v(t));},a,b,n);
}
guide3[] graph(picture pic=currentpicture, triple v(real), real a, real b,
int n=ngraph, bool3 cond(real), interpolate3 join=operator --)
{
return graph(join,cond)(new triple(real t) {
return Scale(pic,v(t));
},a,b,n);
}
guide3 graph(picture pic=currentpicture, triple[] v,
interpolate3 join=operator --)
{
int i=0;
return graph(join)(new triple(real) {
triple w=Scale(pic,v[i]);
++i;
return w;
},0,0,v.length-1);
}
guide3[] graph(picture pic=currentpicture, triple[] v, bool3[] cond,
interpolate3 join=operator --)
{
int n=v.length;
int i=0;
triple w;
checkconditionlength(cond.length,n);
bool3 condition(real) {
bool b=cond[i];
if(b) w=Scale(pic,v[i]);
++i;
return b;
}
return graph(join,condition)(new triple(real) {return w;},0,0,n-1);
}
guide3 graph(picture pic=currentpicture, real[] x, real[] y, real[] z,
interpolate3 join=operator --)
{
int n=x.length;
checklengths(n,y.length);
checklengths(n,z.length);
int i=0;
return graph(join)(new triple(real) {
triple w=Scale(pic,(x[i],y[i],z[i]));
++i;
return w;
},0,0,n-1);
}
guide3[] graph(picture pic=currentpicture, real[] x, real[] y, real[] z,
bool3[] cond, interpolate3 join=operator --)
{
int n=x.length;
checklengths(n,y.length);
checklengths(n,z.length);
int i=0;
triple w;
checkconditionlength(cond.length,n);
bool3 condition(real) {
bool3 b=cond[i];
if(b != false) w=Scale(pic,(x[i],y[i],z[i]));
++i;
return b;
}
return graph(join,condition)(new triple(real) {return w;},0,0,n-1);
}
// The graph of a function along a path.
guide3 graph(triple F(path, real), path p, int n=1,
interpolate3 join=operator --)
{
guide3 g=join(...sequence(new guide3(int i) {
return F(p,i/n);
},n*length(p)));
return cyclic(p) ? join(g,cycle) : join(g,F(p,length(p)));
}
guide3 graph(triple F(pair), path p, int n=1, interpolate3 join=operator --)
{
return graph(new triple(path p, real position)
{return F(point(p,position));},p,n,join);
}
guide3 graph(picture pic=currentpicture, real f(pair), path p, int n=1,
interpolate3 join=operator --)
{
return graph(new triple(pair z) {return Scale(pic,(z.x,z.y,f(z)));},p,n,
join);
}
guide3 graph(real f(pair), path p, int n=1, real T(pair),
interpolate3 join=operator --)
{
return graph(new triple(pair z) {pair w=T(z); return (w.x,w.y,f(w));},p,n,
join);
}
// Connect points in v into segments corresponding to consecutive true elements
// of b using interpolation operator join.
path3[] segment(triple[] v, bool[] cond, interpolate3 join=operator --)
{
checkconditionlength(cond.length,v.length);
int[][] segment=segment(cond);
return sequence(new path3(int i) {return join(...v[segment[i]]);},
segment.length);
}
bool uperiodic(triple[][] a) {
int n=a.length;
if(n == 0) return false;
int m=a[0].length;
triple[] a0=a[0];
triple[] a1=a[n-1];
real epsilon=sqrtEpsilon*norm(a);
for(int j=0; j < m; ++j)
if(abs(a0[j]-a1[j]) > epsilon) return false;
return true;
}
bool vperiodic(triple[][] a) {
int n=a.length;
if(n == 0) return false;
int m=a[0].length-1;
real epsilon=sqrtEpsilon*norm(a);
for(int i=0; i < n; ++i)
if(abs(a[i][0]-a[i][m]) > epsilon) return false;
return true;
}
// return the surface described by a matrix f
surface surface(triple[][] f, bool[][] cond={})
{
if(!rectangular(f)) abort("matrix is not rectangular");
int nx=f.length-1;
int ny=nx > 0 ? f[0].length-1 : 0;
bool all=cond.length == 0;
int count;
if(all)
count=nx*ny;
else {
count=0;
for(int i=0; i < nx; ++i) {
bool[] condi=cond[i];
bool[] condp=cond[i+1];
for(int j=0; j < ny; ++j)
if(condi[j] && condi[j+1] && condp[j] && condp[j+1]) ++count;
}
}
surface s=surface(count);
s.index=new int[nx][ny];
int k=-1;
for(int i=0; i < nx; ++i) {
bool[] condi,condp;
if(!all) {
condi=cond[i];
condp=cond[i+1];
}
triple[] fi=f[i];
triple[] fp=f[i+1];
int[] indexi=s.index[i];
for(int j=0; j < ny; ++j) {
if(all || (condi[j] && condi[j+1] && condp[j] && condp[j+1]))
s.s[++k]=patch(new triple[] {fi[j],fp[j],fp[j+1],fi[j+1]});
indexi[j]=k;
}
}
if(count == nx*ny) {
if(uperiodic(f)) s.ucyclic(true);
if(vperiodic(f)) s.vcyclic(true);
}
return s;
}
surface bispline(real[][] z, real[][] p, real[][] q, real[][] r,
real[] x, real[] y, bool[][] cond={})
{ // z[i][j] is the value at (x[i],y[j])
// p and q are the first derivatives with respect to x and y, respectively
// r is the second derivative ddu/dxdy
int n=x.length-1;
int m=y.length-1;
bool all=cond.length == 0;
int count;
if(all)
count=n*m;
else {
count=0;
for(int i=0; i < n; ++i) {
bool[] condi=cond[i];
for(int j=0; j < m; ++j)
if(condi[j]) ++count;
}
}
surface s=surface(count);
s.index=new int[n][m];
int k=0;
for(int i=0; i < n; ++i) {
int ip=i+1;
real xi=x[i];
real xp=x[ip];
real x1=interp(xi,xp,1/3);
real x2=interp(xi,xp,2/3);
real hx=x1-xi;
real[] zi=z[i];
real[] zp=z[ip];
real[] ri=r[i];
real[] rp=r[ip];
real[] pi=p[i];
real[] pp=p[ip];
real[] qi=q[i];
real[] qp=q[ip];
int[] indexi=s.index[i];
bool[] condi=all ? null : cond[i];
for(int j=0; j < m; ++j) {
if(all || condi[j]) {
real yj=y[j];
int jp=j+1;
real yp=y[jp];
real y1=interp(yj,yp,1/3);
real y2=interp(yj,yp,2/3);
real hy=y1-yj;
real hxy=hx*hy;
real zij=zi[j];
real zip=zi[jp];
real zpj=zp[j];
real zpp=zp[jp];
real pij=hx*pi[j];
real ppj=hx*pp[j];
real qip=hy*qi[jp];
real qpp=hy*qp[jp];
real zippip=zip+hx*pi[jp];
real zppmppp=zpp-hx*pp[jp];
real zijqij=zij+hy*qi[j];
real zpjqpj=zpj+hy*qp[j];
s.s[k]=patch(new triple[][] {
{(xi,yj,zij),(xi,y1,zijqij),(xi,y2,zip-qip),(xi,yp,zip)},
{(x1,yj,zij+pij),(x1,y1,zijqij+pij+hxy*ri[j]),
(x1,y2,zippip-qip-hxy*ri[jp]),(x1,yp,zippip)},
{(x2,yj,zpj-ppj),(x2,y1,zpjqpj-ppj-hxy*rp[j]),
(x2,y2,zppmppp-qpp+hxy*rp[jp]),(x2,yp,zppmppp)},
{(xp,yj,zpj),(xp,y1,zpjqpj),(xp,y2,zpp-qpp),(xp,yp,zpp)}},copy=false);
indexi[j]=k;
++k;
}
}
}
return s;
}
// return the surface described by a real matrix f, interpolated with
// xsplinetype and ysplinetype.
surface surface(real[][] f, real[] x, real[] y,
splinetype xsplinetype=null, splinetype ysplinetype=xsplinetype,
bool[][] cond={})
{
real epsilon=sqrtEpsilon*norm(y);
if(xsplinetype == null)
xsplinetype=(abs(x[0]-x[x.length-1]) <= epsilon) ? periodic : notaknot;
if(ysplinetype == null)
ysplinetype=(abs(y[0]-y[y.length-1]) <= epsilon) ? periodic : notaknot;
int n=x.length; int m=y.length;
real[][] ft=transpose(f);
real[][] tp=new real[m][];
for(int j=0; j < m; ++j)
tp[j]=xsplinetype(x,ft[j]);
real[][] q=new real[n][];
for(int i=0; i < n; ++i)
q[i]=ysplinetype(y,f[i]);
real[][] qt=transpose(q);
real[] d1=xsplinetype(x,qt[0]);
real[] d2=xsplinetype(x,qt[m-1]);
real[][] r=new real[n][];
real[][] p=transpose(tp);
for(int i=0; i < n; ++i)
r[i]=clamped(d1[i],d2[i])(y,p[i]);
surface s=bispline(f,p,q,r,x,y,cond);
if(xsplinetype == periodic) s.ucyclic(true);
if(ysplinetype == periodic) s.vcyclic(true);
return s;
}
// return the surface described by a real matrix f, interpolated with
// xsplinetype and ysplinetype.
surface surface(real[][] f, pair a, pair b, splinetype xsplinetype,
splinetype ysplinetype=xsplinetype, bool[][] cond={})
{
if(!rectangular(f)) abort("matrix is not rectangular");
int nx=f.length-1;
int ny=nx > 0 ? f[0].length-1 : 0;
if(nx == 0 || ny == 0) return nullsurface;
real[] x=uniform(a.x,b.x,nx);
real[] y=uniform(a.y,b.y,ny);
return surface(f,x,y,xsplinetype,ysplinetype,cond);
}
// return the surface described by a real matrix f, interpolated linearly.
surface surface(real[][] f, pair a, pair b, bool[][] cond={})
{
if(!rectangular(f)) abort("matrix is not rectangular");
int nx=f.length-1;
int ny=nx > 0 ? f[0].length-1 : 0;
if(nx == 0 || ny == 0) return nullsurface;
bool all=cond.length == 0;
triple[][] v=new triple[nx+1][ny+1];
for(int i=0; i <= nx; ++i) {
real x=interp(a.x,b.x,i/nx);
bool[] condi=all ? null : cond[i];
triple[] vi=v[i];
real[] fi=f[i];
for(int j=0; j <= ny; ++j)
if(all || condi[j])
vi[j]=(x,interp(a.y,b.y,j/ny),fi[j]);
}
return surface(v,cond);
}
// return the surface described by a parametric function f over box(a,b),
// interpolated linearly.
surface surface(triple f(pair z), pair a, pair b, int nu=nmesh, int nv=nu,
bool cond(pair z)=null)
{
if(nu <= 0 || nv <= 0) return nullsurface;
bool[][] active;
bool all=cond == null;
if(!all) active=new bool[nu+1][nv+1];
real du=1/nu;
real dv=1/nv;
pair Idv=(0,dv);
pair dz=(du,dv);
triple[][] v=new triple[nu+1][nv+1];
for(int i=0; i <= nu; ++i) {
real x=interp(a.x,b.x,i*du);
bool[] activei=all ? null : active[i];
triple[] vi=v[i];
for(int j=0; j <= nv; ++j) {
pair z=(x,interp(a.y,b.y,j*dv));
if(all || (activei[j]=cond(z))) vi[j]=f(z);
}
}
return surface(v,active);
}
// return the surface described by a parametric function f over box(a,b),
// interpolated with usplinetype and vsplinetype.
surface surface(triple f(pair z), pair a, pair b, int nu=nmesh, int nv=nu,
splinetype[] usplinetype, splinetype[] vsplinetype=Spline,
bool cond(pair z)=null)
{
return surface(f,uniform(a.x,b.x,nu),uniform(a.y,b.y,nv),
usplinetype,vsplinetype,cond);
}
// return the surface described by a real function f over box(a,b),
// interpolated linearly.
surface surface(real f(pair z), pair a, pair b, int nx=nmesh, int ny=nx,
bool cond(pair z)=null)
{
return surface(new triple(pair z) {return (z.x,z.y,f(z));},a,b,nx,ny,cond);
}
// return the surface described by a real function f over box(a,b),
// interpolated with xsplinetype and ysplinetype.
surface surface(real f(pair z), pair a, pair b, int nx=nmesh, int ny=nx,
splinetype xsplinetype, splinetype ysplinetype=xsplinetype,
bool cond(pair z)=null)
{
bool[][] active;
bool all=cond == null;
if(!all) active=new bool[nx+1][ny+1];
real dx=1/nx;
real dy=1/ny;
pair Idy=(0,dy);
pair dz=(dx,dy);
real[][] F=new real[nx+1][ny+1];
real[] x=uniform(a.x,b.x,nx);
real[] y=uniform(a.y,b.y,ny);
for(int i=0; i <= nx; ++i) {
bool[] activei=all ? null : active[i];
real[] Fi=F[i];
real x=x[i];
for(int j=0; j <= ny; ++j) {
pair z=(x,y[j]);
Fi[j]=f(z);
if(!all) activei[j]=cond(z);
}
}
return surface(F,x,y,xsplinetype,ysplinetype,active);
}
guide3[][] lift(real f(real x, real y), guide[][] g,
interpolate3 join=operator --)
{
guide3[][] G=new guide3[g.length][];
for(int cnt=0; cnt < g.length; ++cnt) {
guide[] gcnt=g[cnt];
guide3[] Gcnt=new guide3[gcnt.length];
for(int i=0; i < gcnt.length; ++i) {
guide gcnti=gcnt[i];
guide3 Gcnti=join(...sequence(new guide3(int j) {
pair z=point(gcnti,j);
return (z.x,z.y,f(z.x,z.y));
},size(gcnti)));
if(cyclic(gcnti)) Gcnti=Gcnti..cycle;
Gcnt[i]=Gcnti;
}
G[cnt]=Gcnt;
}
return G;
}
guide3[][] lift(real f(pair z), guide[][] g, interpolate3 join=operator --)
{
return lift(new real(real x, real y) {return f((x,y));},g,join);
}
void draw(picture pic=currentpicture, Label[] L=new Label[],
guide3[][] g, pen[] p, light light=currentlight, string name="",
render render=defaultrender,
interaction interaction=LabelInteraction())
{
pen thin=is3D() ? thin() : defaultpen;
bool group=g.length > 1 && (name != "" || render.defaultnames);
if(group)
begingroup3(pic,name == "" ? "contours" : name,render);
for(int cnt=0; cnt < g.length; ++cnt) {
guide3[] gcnt=g[cnt];
pen pcnt=thin+p[cnt];
for(int i=0; i < gcnt.length; ++i)
draw(pic,gcnt[i],pcnt,light,name);
if(L.length > 0) {
Label Lcnt=L[cnt];
for(int i=0; i < gcnt.length; ++i) {
if(Lcnt.s != "" && size(gcnt[i]) > 1)
label(pic,Lcnt,gcnt[i],pcnt,name,interaction);
}
}
}
if(group)
endgroup3(pic);
}
void draw(picture pic=currentpicture, Label[] L=new Label[],
guide3[][] g, pen p=currentpen, light light=currentlight,
string name="", render render=defaultrender,
interaction interaction=LabelInteraction())
{
draw(pic,L,g,sequence(new pen(int) {return p;},g.length),light,name,
render,interaction);
}
real maxlength(triple f(pair z), pair a, pair b, int nu, int nv)
{
return min(abs(f((b.x,a.y))-f(a))/nu,abs(f((a.x,b.y))-f(a))/nv);
}
// return a vector field on a parametric surface f over box(a,b).
picture vectorfield(path3 vector(pair v), triple f(pair z), pair a, pair b,
int nu=nmesh, int nv=nu, bool truesize=false,
real maxlength=truesize ? 0 : maxlength(f,a,b,nu,nv),
bool cond(pair z)=null, pen p=currentpen,
arrowbar3 arrow=Arrow3, margin3 margin=PenMargin3,
string name="", render render=defaultrender)
{
picture pic;
real du=1/nu;
real dv=1/nv;
bool all=cond == null;
real scale;
if(maxlength > 0) {
real size(pair z) {
path3 g=vector(z);
return abs(point(g,size(g)-1)-point(g,0));
}
real max=size((0,0));
for(int i=0; i <= nu; ++i) {
real x=interp(a.x,b.x,i*du);
for(int j=0; j <= nv; ++j)
max=max(max,size((x,interp(a.y,b.y,j*dv))));
}
scale=max > 0 ? maxlength/max : 1;
} else scale=1;
bool group=name != "" || render.defaultnames;
if(group)
begingroup3(pic,name == "" ? "vectorfield" : name,render);
for(int i=0; i <= nu; ++i) {
real x=interp(a.x,b.x,i*du);
for(int j=0; j <= nv; ++j) {
pair z=(x,interp(a.y,b.y,j*dv));
if(all || cond(z)) {
path3 g=scale3(scale)*vector(z);
string name="vector";
if(truesize) {
picture opic;
draw(opic,g,p,arrow,margin,name,render);
add(pic,opic,f(z));
} else
draw(pic,shift(f(z))*g,p,arrow,margin,name,render);
}
}
}
if(group)
endgroup3(pic);
return pic;
}
triple polar(real r, real theta, real phi)
{
return r*expi(theta,phi);
}
guide3 polargraph(real r(real,real), real theta(real), real phi(real),
int n=ngraph, interpolate3 join=operator --)
{
return graph(join)(new triple(real t) {
return polar(r(theta(t),phi(t)),theta(t),phi(t));
},0,1,n);
}
// True arc
path3 Arc(triple c, triple v1, triple v2, triple normal=O, bool direction=CCW,
int n=nCircle)
{
v1 -= c;
real r=abs(v1);
v1=unit(v1);
v2=unit(v2-c);
if(normal == O) {
normal=cross(v1,v2);
if(normal == O) abort("explicit normal required for these endpoints");
}
transform3 T=align(unit(normal));
transform3 Tinv=transpose(T);
v1=Tinv*v1;
v2=Tinv*v2;
real fuzz=sqrtEpsilon*max(abs(v1),abs(v2));
if(abs(v1.z) > fuzz || abs(v2.z) > fuzz)
abort("invalid normal vector");
real phi1=radians(longitude(v1,warn=false));
real phi2=radians(longitude(v2,warn=false));
if(direction) {
if(phi1 >= phi2) phi1 -= 2pi;
} else if(phi2 >= phi1) phi2 -= 2pi;
static real piby2=pi/2;
return shift(c)*T*polargraph(new real(real theta, real phi) {return r;},
new real(real t) {return piby2;},
new real(real t) {return interp(phi1,phi2,t);},
n,operator ..);
}
path3 Arc(triple c, real r, real theta1, real phi1, real theta2, real phi2,
triple normal=O, bool direction, int n=nCircle)
{
return Arc(c,c+r*dir(theta1,phi1),c+r*dir(theta2,phi2),normal,direction,n);
}
path3 Arc(triple c, real r, real theta1, real phi1, real theta2, real phi2,
triple normal=O, int n=nCircle)
{
return Arc(c,r,theta1,phi1,theta2,phi2,normal,
theta2 > theta1 || (theta2 == theta1 && phi2 >= phi1) ? CCW : CW,
n);
}
// True circle
path3 Circle(triple c, real r, triple normal=Z, int n=nCircle)
{
static real piby2=pi/2;
return shift(c)*align(unit(normal))*
polargraph(new real(real theta, real phi) {return r;},
new real(real t) {return piby2;},
new real(real t) {return interp(0,2pi,t);},n,operator ..);
}